Properties

Label 2-150-25.11-c5-0-22
Degree $2$
Conductor $150$
Sign $-0.963 + 0.269i$
Analytic cond. $24.0575$
Root an. cond. $4.90485$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.23 − 2.35i)2-s + (2.78 − 8.55i)3-s + (4.94 − 15.2i)4-s + (−21.1 − 51.7i)5-s + (−11.1 − 34.2i)6-s + 92.5·7-s + (−19.7 − 60.8i)8-s + (−65.5 − 47.6i)9-s + (−190. − 117. i)10-s + (377. − 274. i)11-s + (−116. − 84.6i)12-s + (−257. − 187. i)13-s + (299. − 217. i)14-s + (−501. + 36.8i)15-s + (−207. − 150. i)16-s + (27.7 + 85.5i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (−0.377 − 0.925i)5-s + (−0.126 − 0.388i)6-s + 0.714·7-s + (−0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s + (−0.600 − 0.372i)10-s + (0.940 − 0.683i)11-s + (−0.233 − 0.169i)12-s + (−0.422 − 0.307i)13-s + (0.408 − 0.296i)14-s + (−0.575 + 0.0423i)15-s + (−0.202 − 0.146i)16-s + (0.0233 + 0.0718i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.269i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.963 + 0.269i$
Analytic conductor: \(24.0575\)
Root analytic conductor: \(4.90485\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :5/2),\ -0.963 + 0.269i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.444935375\)
\(L(\frac12)\) \(\approx\) \(2.444935375\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.23 + 2.35i)T \)
3 \( 1 + (-2.78 + 8.55i)T \)
5 \( 1 + (21.1 + 51.7i)T \)
good7 \( 1 - 92.5T + 1.68e4T^{2} \)
11 \( 1 + (-377. + 274. i)T + (4.97e4 - 1.53e5i)T^{2} \)
13 \( 1 + (257. + 187. i)T + (1.14e5 + 3.53e5i)T^{2} \)
17 \( 1 + (-27.7 - 85.5i)T + (-1.14e6 + 8.34e5i)T^{2} \)
19 \( 1 + (361. + 1.11e3i)T + (-2.00e6 + 1.45e6i)T^{2} \)
23 \( 1 + (1.86e3 - 1.35e3i)T + (1.98e6 - 6.12e6i)T^{2} \)
29 \( 1 + (1.47e3 - 4.54e3i)T + (-1.65e7 - 1.20e7i)T^{2} \)
31 \( 1 + (1.74e3 + 5.36e3i)T + (-2.31e7 + 1.68e7i)T^{2} \)
37 \( 1 + (691. + 502. i)T + (2.14e7 + 6.59e7i)T^{2} \)
41 \( 1 + (-9.80e3 - 7.12e3i)T + (3.58e7 + 1.10e8i)T^{2} \)
43 \( 1 - 1.04e3T + 1.47e8T^{2} \)
47 \( 1 + (-5.46e3 + 1.68e4i)T + (-1.85e8 - 1.34e8i)T^{2} \)
53 \( 1 + (-58.8 + 181. i)T + (-3.38e8 - 2.45e8i)T^{2} \)
59 \( 1 + (5.23e3 + 3.80e3i)T + (2.20e8 + 6.79e8i)T^{2} \)
61 \( 1 + (-3.64e3 + 2.64e3i)T + (2.60e8 - 8.03e8i)T^{2} \)
67 \( 1 + (-3.00e3 - 9.24e3i)T + (-1.09e9 + 7.93e8i)T^{2} \)
71 \( 1 + (-5.06e3 + 1.55e4i)T + (-1.45e9 - 1.06e9i)T^{2} \)
73 \( 1 + (-4.62e4 + 3.36e4i)T + (6.40e8 - 1.97e9i)T^{2} \)
79 \( 1 + (-1.30e4 + 4.00e4i)T + (-2.48e9 - 1.80e9i)T^{2} \)
83 \( 1 + (-1.62e4 - 5.01e4i)T + (-3.18e9 + 2.31e9i)T^{2} \)
89 \( 1 + (-5.14e4 + 3.73e4i)T + (1.72e9 - 5.31e9i)T^{2} \)
97 \( 1 + (-2.54e4 + 7.82e4i)T + (-6.94e9 - 5.04e9i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.76163059215100986924719003528, −11.09204824334686430304072007100, −9.464572912815639090131415320401, −8.510032986472553508685593666894, −7.44167411954464307244699840690, −5.97435966110842002188776252282, −4.81999815095408926610485930773, −3.61417976644315035584117388753, −1.87093699280759030560895337508, −0.66326310107304404171473116396, 2.18915755917880858009153089397, 3.74988775725062912150085470250, 4.58965401479952852132088352263, 6.09932214717881285494340125208, 7.21916753931698439245128431688, 8.176905845026589844753538019658, 9.531197404896221208381700495948, 10.66121225267718440226692276777, 11.63937502104500828065727491821, 12.42273751221826217495864012489

Graph of the $Z$-function along the critical line